# Zero Matrix

A zero matrix is a matrix that has all its elements equal to zero. Since a zero matrix contains only zeros as its elements, therefore, it is also called a null matrix. A zero matrix can be a square matrix.

A zero matrix is denoted by ‘O’. It is an additive identity matrix that results in the same matrix when added to a matrix of order m x n.

## Definition of Zero Matrix

A zero matrix is the arrangement of zero elements into rows and columns. A zero matrix is a matrix with all its entries equal to zero. It is denoted by ‘O’ that can be expressed with a subscript to represent the dimension of the matrix.

$$\begin{array}{l}0_{m, n}=\left[\begin{array}{cccc} 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{array}\right]_{m \times n}\end{array}$$

## Examples of Zero Matrices

The zero matrices of the different orders are given below:

• Zero matrix of order 1 x 1 → A1,1 = [0]
• Zero matrix of order 1 x 2 → A1,2 = [0, 0]
• $$\begin{array}{l}\text{Zero matrix of order 2 x 1 → }A_{2,1} = \begin{bmatrix} 0\\ 0 \end{bmatrix}\end{array}$$
• $$\begin{array}{l}\text{Zero matrix of order 2 x 2 → } A_{2,2} = \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}\end{array}$$
• $$\begin{array}{l}\text{Zero matrix of order 3 x 3 → }A_{3,3 }= \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{array}$$
Facts:

1. The entries in a zero matrix are equal to zero.
2. When we add a matrix, say A, to a zero matrix, then the resulting matrix will be A only.

## Types of Matrices

There are majorly seven different types of matrices. They are:

1. Row matrix
2. Column matrix
3. Square matrix
4. Diagonal matrix
5. Scalar matrix
6. Identity matrix
7. Zero matrix

When a non zero matrix of order m x n, is added with a zero matrix of the same order, then the sum will be the original matrix.

Suppose, A = [aij] is an m × n matrix and O is an m × n zero matrix, then;

A + O = O + A = A

### Examples

• $$\begin{array}{l}\left[\begin{array}{ll} 1 & 3\\ 2 & 4 \end{array}\right]+\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]\end{array}$$
• $$\begin{array}{l}\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right]+\left[\begin{array}{rr} -1 & 2 \\ 5 & 8 \\ -7 & 9 \end{array}\right]=\left[\begin{array}{cc} -1 & 2 \\ 5 & 8 \\ -7 & 9 \end{array}\right]\end{array}$$

We can see, in the above examples, the zero matrix when added to another matrix, does not change the identity of the matrix. Therefore, it is called the additive identity for the matrix addition.

## Zero Matrix – Product of two matrices

If two non-zero matrices are multiplied together, then it is possible to get a zero matrix.

We can say, for two real numbers, say x and y, if xy = 0, then either x = 0 or y = 0. The same concept is applicable for matrices too.

If the rows of a matrix A have zero elements and columns of matrix B (same order) have zero elements, then the product of matrices A and B will result in a zero matrix. Let us see an example.

Example:

$$\begin{array}{l}If~ A = \left [\begin{array}{ll} 0 & 0 \\ 3 & 7 \end{array}\right]and ~B= \left[\begin{array}{rr} 0 & 2 \\ 0 & 8 \end{array}\right]~then ~AB ~is ~equal ~to:\end{array}$$
$$\begin{array}{l}AB = \left[\begin{array}{ll} 0 & 0 \\ 3 & 7 \end{array}\right]\times \left[\begin{array}{rr} 0 & 2 \\ 0 & 8 \end{array}\right]=\left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right]\end{array}$$

Note: If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

## Frequently Asked Questions – FAQs

Q1

### What is a zero matrix? Give an example.

A zero matrix is a matrix that contains all the elements equal to 0. It is denoted by the symbol ‘O’. For example, [0]1×1 is a zero matrix of order 1.

Q2

### What is the order of the zero matrix?

The order of the zero matrix is m x n, where m is number of rows and n is number of columns.

Q3

### Is a zero matrix a square matrix?

It is not necessary that a square matrix is always a square matrix.

Q4

### Is zero matrix a diagonal matrix?

A diagonal matrix has non-zero elements on its diagonal whereas the off-diagonals elements are zero. But a zero matrix has all its elements as zero.

Q5

### What is the result of the sum of a non-zero matrix with a zero matrix?

The sum of a non-zero matrix, say A, with a zero matrix, O, will result in the matrix A, itself. A + O = A = O + A